Gap conditions for representations of symmetric groups

Representations of symmetric groups

In this thesis we study the ordinary and the modular representation theory of the symmetric group. In particular we focus our work on different important open questions in the area. 1. Foulkes’ Conjecture In Chapter 2 we focus our attention on the long standing open problem known as Foulkes’ Conjecture. We use methods from character theory of symmetric groups to determine new information on the...

Representations of Symmetric Groups

A surprising theorem in the modular representation theory of symmetric groups uses induction and restriction functors to define an action of an affine Kac-Moody special linear algebra on the level of Grothendieck groups. This action identifies the direct sum of Grothendieck groups with an integrable highest weight module of the Kac-Moody algebra. The purpose of this write-up is to provide a gen...

Symmetric Representations of Knot Groups

In this paper we give an explicit and constructive description of the pairs (p. A) of elements in the symmetric group S, which can be realized as the image of the meridian-longitude pair of some knot K in the 3-sphere S3 under a representation xi (S’ K) -t S,,. The result is applied to give an otherwise nonobvious restriction on the numbers of branch curves of a branched covering , of S’, answe...

Modular Representations of Symmetric Groups

Over the last few weeks, we have seen lots of things in representation theory—we began by looking at an onslaught of examples to see explicitly what representations look like, we discussed character theory and looked at character tables, we talked about representations from a module-theoretic perspective, and then we saw how we can model the representation theory of the symmetric group via the ...

Projective Representations of Generalized Symmetric Groups